Fixed-order PCA: Theory for Overestimated Factor Models

TL;DR

This paper develops asymptotic theory for fixed-order PCA in high-dimensional factor models, showing that extra eigencomponents are noise-driven and establishing factor estimation consistency.

Contribution

It provides a theoretical foundation for using a conservative upper bound on the number of factors in PCA, relaxing the need for exact dimension selection.

Findings

Extra eigencomponents beyond the true factors are asymptotically noise-driven.

Introduces two rotation maps, H' and H+, for factor estimation.

Proves asymptotic normality of treatment-effect estimates in factor-augmented regression.

Abstract

We develop asymptotic theory for principal component analysis (PCA) of a high-dimensional factor model in which the working dimension $R$ is fixed and only required to satisfy $R \ge r$, where $r$ is the true number of factors. Building on anisotropic local laws from random matrix theory, we show that the ``extra'' empirical eigencomponents beyond the $r$-th are asymptotically noise-governed, incoherent, and nearly orthogonal to the factor loadings. We introduce two rotations, an expanded $r\times R$ map $H'$ and a compressed $R\times r$ map $H^{+}$, and establish consistency of the estimated factors under both. As an application, we analyze a factor-augmented regression for treatment-effect inference and prove $\sqrt{T}$-asymptotic normality for every fixed $R \ge r$. These results provide a theoretical underpinning for the common empirical practice of adopting a conservative upper bound on the number of factors, and shift the analytical burden from consistent dimension selection to the milder requirement of bounding $r$ from above.